Supplementary information for

Graphene hot-electron light bulb: incandescence from hBN-

encapsulated graphene in air

Seok-Kyun Son1,4, Makars Šiškins2,4, Ciaran Mullan2,4, Jun Yin2, Vasyl G. Kravets2, Aleksey

Kozikov1, Servet Ozdemir2, Manal Alhazmi2, Matthew Holwill1, Kenji Watanabe3, Takashi

Taniguchi3, Davit Ghazaryan1, Kostya S. Novoselov1,2, Vladimir I. Fal’ko1,2, Artem Mishchenko1,2,5

1National Graphene Institute, University of Manchester, Manchester M13 9PL, UK

2School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, UK

3National Institute for Materials Science, 1-1 Namiki, Tsukuba, 305-0044, Japan

4 These authors contributed equally

5 e-mail: artem.mishchenko@gmail.com

1. Device performance and stability in high current regime

We have measured more than a dozen of different devices, both standard multi-terminal transport

devices and specifically designed filaments, and most of them survived long-term measurements of at

least several hours at current densities in excess of 108 A/cm2. Naturally, with the higher current

densities the life-time of the devices drops exponentially. As an example, figure S1 shows the stability

of a conventional Hall bar device (hBN/graphene/hBN) in a high current regime – we managed to record

three breakdown curves at ~ 1, 2 and 3·108 A/cm2 current densities, with the life-time of the order of

105, 103, and 101 seconds, respectively.

For the standard Hall bar devices, the main mechanism of failure is most likely the breakdown at the

graphene/gold interface probably due to an increased mobility of gold atoms at elevated temperatures

in the presence of high electric fields, which exposes the interface to the ambient and allows for thermal

oxidation of graphene along the 1D graphene/metal contact. In the specially designed filaments (with a

wide contact area and a long comb shaped contact perimeter) the breakdown usually happens after much

longer periods of time, because the graphene/metal interface is at much lower temperatures thanks to

the special design of the contact (cf figures S1d and S2a). Typically, the device breaks in the middle of

the filament (figure S2) and also at higher current densities (> 3·108 A/cm2). The location of the break

usually can be traced back to the presence of bubbles of trapped contaminants between the graphene

layer and one of the encapsulating hBN crystals. At these high temperatures the content of the bubbles

(mostly hydrocarbons) decomposes producing reactive chemical species which can corrode the

graphene (as seen in figure S2 panel d), or even blast the heterostructure due to the formation of gaseous

products from thermal decomposition of trapped contaminants (figure S2 panel b).

The maximum attainable current density is comparable to non-encapsulated devices measured in

vacuum, 4.5·108 A/cm2 [1], and approximately twice smaller than that of single-walled carbon

nanotubes on SiO2 substrate, 109 A/cm2 [2]. The hBN protection properties were recently exploited for

infrared (~ 4 µm) emitters based on a few (6-8) layer graphene [3]. Encapsulation of the few-layer

graphene heating element with 13 nm hBN allowed extended operation (> 1000 hours) in ambient

conditions. The device was Joule-heated (current density < 107 A/cm2) to 440-530 K using AC current.

Thus, the long-term stability of hBN encapsulated devices depends on several factors, such as the

presence of trapped air-born hydrocarbons in the form of bubbles during the device fabrication, the

current density, the contact resistance, etc.

Figure S1. Long-term stability of standard Hall bar devices (encapsulated hBN/graphene/hBN

heterostructures) in a high current regime in ambient conditions.

0 50 100 1500

1

2

3

Cu

rre

nt

de

nsity (

10

8 A

/cm

2)

Time (h)

~ 140 h

0 40 80 1200

1

2

3

Cu

rre

nt

de

nsity (

10

8 A

/cm

2)

Time (min)

~ 70 min

~ 110 min

0 15 30 450

1

2

3

Cu

rre

nt

de

nsity (

10

8 A

/cm

2)

Time (s)

~ 40 s

(a) (b)

(c)

1 2

3

(d)

2

3

1

5 µm

Figure S2. AFM images of hBN-encapsulated graphene filaments after total (panels a and b) and partial

(panels c and d) breakdown. Device on panels a and b has burned after glowing for ~ 12 hours at ~

2·108 A/cm2 current density. Device on panels c and d (the same device as in the figure 2, main text)

developed a blister when operated at ~ 3·108 A/cm2 for a couple of hours. Raman measurements confirm

that graphene is partially decomposed: intensity of 2D peak in the blister area is 10 times smaller as

compared to the rest of the device. Nevertheless, even after the formation of blister and partial

breakdown of the graphene the device was still operational and survived many high current regime

measurements afterwards. The corrugated features present in device channel area (most notable on the

panel d) are the polymer residues on top of hBN left after the device fabrication.

2. Extracting temperatures from grey-body radiation and Fabry-Perot cavity models

Spectra of thermal radiation for an hBN-encapsulated graphene can be approximated by Planck’s law,

modified by emissivity ϵ (grey-body radiation):

𝐼𝑔𝑟𝑒𝑦−𝑏𝑜𝑑𝑦(ℎ𝜈, 𝑇) = 𝜖2(ℎ𝜈)3

ℎ2𝑐2

1

exp (ℎ𝜈

𝑘𝐵𝑇) − 1

Following Kirchhoff’s law of thermal radiation we assume the emissivity of monolayer graphene from

its absorptivity ϵ = α ≈ 0.023, which, in the first approximation, is energy-independent in a spectral

range from visible to near infrared [4]. To fit the experimental data, we also used an additional

proportionality coefficient, k, to grey-body model to normalize on arbitrary units of spectra acquired by

Si-based CCD detector: 𝐼𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑(ℎ𝜈, 𝑇) = 𝑘 × 𝐼𝑔𝑟𝑒𝑦−𝑏𝑜𝑑𝑦(ℎ𝜈, 𝑇). In order to simplify the fitting

60nm

5µm 500nm

(a)

(c) (d)

5µm

100nm 8nm

500nm

180nm

(b)

equation, we hereby absorbed all multiplicative constants as a single proportionality coefficient, 𝐾 =

2𝑘𝜖

ℎ2𝑐2, so that 𝐼𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑(ℎ𝜈, 𝑇) = 𝐾 × (ℎ𝜈)3 (exp (ℎ𝜈

𝑘𝐵𝑇) − 1)

−1. Subsequently, we deducted K from

the fit of the highest-intensity spectrum (acquired at 16 V) and then fixed this value, simplifying the

fitting procedure to a single parameter (temperature, T) fit for all subsequent iterations. As seen in figure

S3, panels a and b, for hBN/graphene/hBN heterostructures on quartz substrate light emission spectra

closely follow the grey-body model.

In contrast, when SiO2/Si substrate is used, a photonic cavity formed between hBN/air and SiO2/Si

interfaces shifts the emission spectra to the visible range. Consequently, for the following case we

considered the incandescence model modulated by a spectral line shape of the second mode of the cavity,

using a Lorentzian:

𝐼𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑(ℎ𝜈, 𝑇) = 𝐾 ×(ℎ𝜈)3

exp (ℎ𝜈

𝑘𝐵𝑇) − 1

×(ℏ 𝜏⁄ )2

4(ℎ𝜈 − 2ℎ 𝑡𝑅𝑇⁄ )2 + (ℏ 𝜏⁄ )2

Since both photon decay time, 𝜏 and round trip time, 𝑡𝑅𝑇 could be estimated from known dimensions

of the cavity, model significantly simplifies to have a temperature as the only fitting parameter (see

figure S3, panels c and d).

Figure S3. Thermal radiation of Joule-heated hBN/graphene/hBN devices. (a) Thermal emission

spectra from the quartz-supported device at different bias voltages (from 8 V to 16 V, grey to orange).

The inset shows schematics of light propagation inside the heterostructure. (b) Temperatures deducted

from the fit. (c) Spectra from the SiO2/Si-supported device at different bias (from 9 V to 23 V, violet to

dark red). The inset: schematics of Fabry-Perot cavity formed. (d) Temperature values extracted from

the fit.

Remarkably, this approach provides a fair estimate of sample temperatures under a range of applied

electric fields. Several characteristics of the spectra, however, such as unexpected shoulder seen at ~

1.3 eV in figure S3c, remained yet unclear and required a more sophisticated analysis for a better

understanding of the effect of photonic cavity. We described a more involved model of this particular

case in details in Section 4.

3. Incandescence efficiency – thermal radiation vs electrical power

A supplied electrical power Pelectrical = I·V dissipates as heat, which raises the temperature of a device

(figure S4a). The total radiated power can be approximated from the temperature of incandescence

using Stefan-Boltzmann law, Poptical = AϵσT4 (figure S4b). The efficiency of light emission (Poptical /

Pelectrical) of our devices was in the range of 1–3·10-5, therefore only a small fraction of supplied electrical

power is radiated away. This suggests that hBN encapsulation, substrate, and the metal contacts sink

most of the released heat. When directly comparing with commercial incandescent lamps, the efficiency

of our graphene-based filaments is ~ 2·10-5, while commercial light-bulbs reach values about 2·10-2 [5].

This would correspond to three orders of magnitude better radiation efficiency for conventional

macroscopic light-sources. However, we believe such comparison of macroscopic alternatives with the

2D microscopic light emitters may appear ambiguous and shall be better compared with other known

sources of a more commensurate scale. In figure S4c we compare the efficiency of thermal radiation of

our devices with those reported in literature. The efficiency of our devices was an order of magnitude

better than that for graphene on SiO2/Si substrate [6, 7], and comparable to hBN encapsulated multilayer

graphene thermal emitters [3]. By suspending the graphene filament, the efficiency can be increased by

2–3 orders of magnitude [8], which is due to diminished heat transfer to the substrate. The reason for

better efficiency of hBN encapsulated devices (compared to graphene-on-SiO2 devices) remains unclear

and requires further investigations.

Figure S4. Comparison of attainable temperatures (panel a), optical power (panel b) and emission

efficiency (panel c) versus supplied electrical power.

4. Thermal radiation in a photonic cavity

Here we consider the following 5-layer structure (figure S5) for a black-body-like thermal radiation

emitter. The emitter is represented as the thin layer of hBN dielectric of thickness d1 = 30 nm (medium

1), followed by a graphene monolayer, d2 = 0.34 nm (medium 2), then another thin layer of hBN

dielectric of thickness d3 = 50 nm (medium 3), then a SiO2 dielectric of thickness d4 = 290 nm (medium

4) on a semi-infinite doped silicon substrate (Si).

In our modelling, we estimate the emission from graphene monolayer as the standard black-body

radiation described by Planck’s formula. Following Landau and Lifshitz approach [9] the radiation

energy intensity U(,T) (power per unit area at unit frequency interval) can be expressed as [10]:

𝑈(𝜔, 𝑇) =ℏ𝜔

𝑒ℏ𝜔 𝑘𝐵𝑇⁄ −1

𝑛𝑔𝑟2 𝜔2

(2𝜋)3𝑐2, (1)

where T is a black-body thermal equilibrium temperature, ħ and kB are the reduced Planck’s constant

and the Boltzmann’s constant, respectively, ngr is the real part of the graphene refractive index, c is the

speed of light, and is spectral radiation frequency.

(a)

(c)

(d)

0.0 0.5 1.0 1.50

1000

2000

substrate:

quartz

SiO2/Si

T (

K)

Pelectrical

(MW/m2)

0 500 1000 15000

20

40

60

substrate:

quartz

SiO2/Si

Po

ptica

l (kW

/m2)

Pelectrical

(MW/m2)

107

108

109

1010

10-6

10-5

10-4

10-3

10-2

Freitag, 2010390K 530K

Berciaud, 2010

Kim, 2015

Barnard, 2016this work

1145K 2040K

2350K

1400K 2500K

Po

ptica

l/Pe

lectr

ica

l

Pelectrical

(W/m2)

2800K

440K530K

Kim, 2015

We consider graphene as the source of black-body radiation at a given temperature, T. The emitted

radiation is reflected by the photonic cavity, and therefore the radiation leaving the very top surface of

the heterostructure consists of directly emitted and reflected components. In the case of an isotropic

black-body-like emitter, we have a contribution in measured intensity from light emission directly from

the top surface forwarded to the photodetector and the part modified by the photonic cavity.

The intensity of the emission from the graphene surface (figure S5) is determined by the integration of

all the radiation rays originated from it, by analogy with a radiative heat transfer [11, 12]:

𝐼(𝜔) = ∫ 𝑈(𝜔, 𝑇)𝑆(𝜔, 𝛽)𝛽𝑑𝛽, (2)

where the Planck’s black-body distribution function U(,T) is weighted by S(,) over wavevectors

. The weighting function includes reflections from interfaces for both p- and s-polarised light:

𝑆(𝜔, 𝛽) = 0.5 (𝑅𝑝(𝜔, 𝛽) + 𝑅𝑠(𝜔, 𝛽)), (3)

where 𝑅𝑝,𝑠(𝜔, 𝛽) = |𝑟𝑝,𝑠(𝜔, 𝛽)|2 are the p- and s-polarized light reflectivities from the photonic cavity

in a direction perpendicular to interfaces in the heterostructure [13, 14]. In turn, the dispersion relation

for multiple p- and s- polarized waves was derived using the Fresnel amplitude reflection coefficients

[13]:

𝑟𝑝,𝑠(𝜔, 𝛽) =𝑟𝑗,𝑗+1

𝑝,𝑠+𝑟𝑗+1,𝑗+2

𝑝,𝑠exp(2𝑖𝑞𝑗+1𝑑𝑗+1)

1+𝑟𝑗,𝑗+1𝑝,𝑠

𝑟𝑗+1,𝑗+2𝑝,𝑠

exp(2𝑖𝑞𝑗+1𝑑𝑗+1) (4)

Finally, the Fresnel reflection coefficients for p- and s- polarization at the interfaces are given as

𝑟𝑗,𝑗+1𝑠 =

𝑞𝑗−𝑞𝑗+1

𝑞𝑗+𝑞𝑗+1 (5a)

𝑟𝑗,𝑗+1𝑝 =

𝜀𝑗+1(𝜔)𝑞𝑗−𝜀𝑗(𝜔)𝑞𝑗+1

𝜀𝑗+1(𝜔)𝑞𝑗+𝜀𝑗(𝜔)𝑞𝑗+1 (5b)

Here 𝑞𝑗 = √𝜀𝑗(𝜔) 𝜔2 𝑐2⁄ − 𝛽2 and 𝛽 = (𝜔 𝑐⁄ ) sin 𝜃 are the components of the wavevectors kj

perpendicular and parallel to the surface of the layers, j is a layer index.

The dielectric functions 1() = 3(), 2(), 4(), and 5() for top and bottom hBN layers, graphene,

SiO2 and Si, respectively, were used to calculate the Fresnel reflection coefficients Rp,s(,). The optical

constants of SiO2 film and Si substrate were parameterized by the Cauchy and Cauchy-Urbach functions,

respectively [14]. To find Cauchy coefficients we carried out spectroscopic ellipsometry measurements

of SiO2 film (d2 = 290 nm) on top of Si substrate. The hBN refractive index was also modelled by

Cauchy dispersion law using ellipsometric data taken from [6]. We considered only three coefficients

A = 1.97, B = 0.610-3 m-2, C = 1.910-3 m-4 for the perpendicular component of the refractive index

[15]. The corresponding refractive indices of SiO2 and hBN layers show low dispersion throughout the

visible and near infrared spectral range, averaging at nSiO2 = 1.49±0.015 and nhBN = 2.05±0.1.

The dielectric function of a monolayer graphene can be obtained from the optical conductivity [16]:

𝜀(𝜔) = 1 +𝑖𝜎(𝜔)

1𝜋𝜔𝜀0𝑡𝑔𝑟, (6)

where tgr = 0.34 nm is the thickness of graphene, and 0 is the vacuum permittivity. The optical

conductivity 𝜎(𝜔) is dominated by the two terms, the intraband electron-photon scattering processes,

𝜎𝑖𝑛𝑡𝑟𝑎(𝜔), and the direct interband electron transitions, 𝜎𝑖𝑛𝑡𝑒𝑟(𝜔) [16]. These two terms are as follows:

𝜎𝑖𝑛𝑡𝑟𝑎(𝜔) =𝑖𝑒2𝑘𝐵𝑇

𝜋2ℏ2(𝜔+𝑖 𝜏⁄ )ln [2 cosh (

𝐸𝐹

2𝑘𝐵𝑇)] (7a)

𝜎𝑖𝑛𝑡𝑒𝑟(𝜔) =𝑒2

4ℏ{

1

2+

1

𝜋tan−1 (

ℏ𝜔−2𝐸𝐹

2𝑘𝐵𝑇) −

𝑖

2𝜋ln [

(ℏ𝜔−2𝐸𝐹)2

(ℏ𝜔−2𝐸𝐹)2+4(𝑘𝐵𝑇)2]} (7b)

Here, EF is the Fermi level of graphene, which is dependent on the charge carrier density, and is the

carrier momentum relaxation time.

Using equations 4-7 we calculated the polarized reflectivity at the following interfaces: graphene/hBN,

hBN/SiO2 and SiO2/Si, and then computed the total reflection coefficients Rp,s(,). The spectral

intensity of emission, I(), equation 2, was evaluated by the integration over (over all directions at

which emission energy leave a unit area of graphene surface) from 0 to k0 = 2/ (k0 is the wavevector

of photon in the air) or, or alternatively, by transforming the integral to a variable polar angle θ and

integrating over θ between 0 and /2.

Figure S6 shows the computed intensities of the emission from the modelled system for different

temperatures, as a function of the photon energy. For comparison, the experimental spectral emission

is also displayed in figure S6. The modelled (at T = 2400 K) and the experimental (black line) spectra

match quite well (figure S6b). Note that the small feature (shoulder peak) at around 1.4 eV on

experimental curve is also partially reproduced on the theoretical curve (but not on the simple Fabry-

Perot model as in section 2 of the Supplementary, or in the main text). Therefore this feature can be

attributed to the presence of additional interfaces (such as hBN/SiO2) included in the model in this

section, although this would require additional confirmation by measuring a range of devices with

different thickness of hBN and SiO2 dielectric layers.

Figure S5. Schematics of the considered structure consisting of single layer graphene, encapsulating

hBN stacks, SiO2 layer and Si+ substrate.

Figure S6. Computed emission from Si/SiO2 supported hBN/graphene/hBN heterostructure as a

function of photon energy for different temperatures. For the modelling of thermal radiation we used

the structure shown in figure S5.

5. Linewidth of the incandescent spectra from the photonic cavity

It is interesting to compare the linewidth of the light emission spectra from the photonic cavity with the

photon decay time estimated from the known parameters of the cavity. Figure S7 shows the linewidth

extracted from the Lorentzian peak fits for spectra obtained at different bias voltages (and, hence

different electronic temperatures). The linewidth is mostly independent of applied bias, except for the

lowest bias case, where it slightly deviates from the average value. The average linewidth is ≈ 0.5 eV,

in a good agreement with the expected photon lifetime in our photonic cavity.

Si

SiO2

hBN

hBN

290 nm

50 nm

30 nmGr

(a) (b)

1.0 1.5 2.0 2.50

1

2

3

4

Inte

nsity (

a.u

.)

Energy (eV)

exp

1500 K

2000 K

2400 K

3000 K

1.0 1.5 2.0 2.50

1

Inte

nsity (

a.u

.)

Energy (eV)

exp

2400 K

Figure S7. Linewidths of the light emission spectra (extracted from Lorentz peak fits, dotted

curves on panel a) vs bias voltage (panel b).

6. Temperature distribution in hBN/graphene filaments

To characterise the thermal management in our hBN/graphene hot filaments we considered a simple

model [3, 8, 17] based on a 1D heat diffusion equation

∇(𝑘∇𝑇) +𝑝

𝑙𝑤𝑑−

2𝑔

𝑑(𝑇 − 𝑇0) = 0 (8)

Here, T is a function of x coordinate along the graphene channel of length l, width w, and thickness d.

We choose x = 0 to be in the middle of the filament and the metal contacts are located at 𝑥 = ± 𝑙2⁄ .

𝑝

𝑙𝑤≡ 𝑃 is the electric power (Joule heating) per unit area (W·m−2), k is the in-plane thermal conductivity

(W·m−1·K−1) and g is an out-of-plane (interfacial) thermal conductance (W·m−2·K−1). T0 = 300 K is the

ambient temperature of a substrate, contacts and air. In a zeroth approximation, we assume that P is

uniform along the channel, and both k and g are independent of temperature. Then, with standard

boundary conditions, 𝑇(± 𝑙2⁄ ) = 𝑇0, the solution of equation 8 is straightforward:

𝑇(𝑥) = 𝑇0 +𝑃

2𝑔(1 −

cosh(𝑚𝑥)

cosh(𝑚𝑙

2)), (9)

where 𝑚 = √2𝑔

𝑘𝑑. The temperature profiles from equation 9 are summarised in the figure S8, for a range

of k, g, and P. The best match to our data (power vs temperature) was for g ~ 3.5·105 W·m−2·K−1 and k

~ 500–2500 W·m−1·K−1. The overall shape of the temperature profile is rather insensitive to the value

of k, as long as g is above 105. The smaller the k the more flat is the temperature profile. In our

simulations we ignored the temperature dependence of k (usually, k(T) is modelled as 𝑘(𝑇) = 𝑘0 (𝑇0

𝑇)

𝛾,

where 𝑘(𝑇0) = 𝑘0 and γ ~ 1.9 [17]), the main effect of k(T) of this form is to produce a “hot spot” in

the middle of the device, the effect which we did not observe in our case.

(a) (b)

1.5 2.00

10

20

30

Inte

nsity (

a.u

.)

Energy (eV)

9V

11V

13V

15V

17V

19V

21V

23V

bias voltage:

0 5 10 15 20 250.0

0.2

0.4

0.6

0.8

1.0

Lin

ew

idth

(e

V)

Bias voltage (V)

0.5 eV

The estimated thermal conductance g is well above the measured conductance between graphene and

air (2.9·104 W·m−2·K−1), and even exceeds the theoretical limit from the kinetic theory of gases [18]:

𝑔𝑚𝑎𝑥 ≈5

8𝑃𝑎𝑖𝑟√

3𝑘𝐵𝑇

𝑚~105 (Pair is the atmospheric pressure, and m is the molecular weight of air). On

the other hand, the obtained value of g is ~ 20 times smaller than that reported for graphene/hBN, and

approximately two orders of magnitude lower than reported for graphene/SiO2 and graphene/metal

interfaces [19, 20]. We believe the hBN encapsulation provides an efficient heat spreader, which

prevents the formation of the “hot spot” and allows for a higher current densities and electrical power

to be applied to our devices. We would like to note that the effect of heat spreading was recently reported

for hBN films and laminates [21, 22].

Figure S8. Temperature profiles along the graphene filament (x-axis) calculated for different in-plane

(k) and out-of-plane (g) heat transfer contributions.

7. Possible plasmonic emission from the metal contacts

Here we would like to discuss the possible explanations for the presence of the light emission spots at

the tips of the comb shaped gold contacts (as on figure 2b in the main text). The direct thermal emission

due to the local heating at the graphene/gold interface is rather unlikely (although cannot be completely

excluded) due to a very good thermal conductivity of graphene/gold interface [20] and a low contact

resistance of 1D contacts between hBN/graphene/hBN stack and metal electrode [23]. On the other

hand, we are able to account for the bright dots in the vicinity of gold electrodes using a simple cavity

model (inset in the figure S9) which predicts a plasmon enhancement of the thermal emission. We

consider a light propagation in the dielectric cavity under average angle (θ = 45°), because the large

fraction of the emitted light at this angle can be trapped into a photonic cavity and can propagate a

x

T log P

g = 2.5·104

k = 500

k = 2500

k = 5000

g = 0 g = 105 g = 106

relatively long distance. The transmission coefficients for the cavity can be obtained by the following

reasoning: the transmitted field is the sum of the first reflection at the SiO2/Si interface and the second

reflection at the hBN/air interface, and the cavity will collect the light around the metal contacts after

N = L/(4d) consecutive reflections (L ≈ 2-5 µm is the distance between the glowing hot part of the

device and the metal contact, and d is the thickness of the cavity).

The corresponding p-polarised spectra (which can excite a plasmon on a surface of gold electrodes)

propagating along the dielectric cavity are shown in figure S9. Black curve displays the p-polarised

emission spectra, Ep, in the cavity after two consecutive interferences in vicinity of graphene. Red curve

is the intensity of p-polarised thermal radiation in the vicinity of electrodes, which can be estimated as

EpN. Blue curve is the plasmon-enhanced spectrum due to the coupling of the incoming light to the

plasmon mode of gold. In this case the plasmon mode will resonate in the exciting field and enhance

the local electromagnetic field by a factor Q ~ Re(ε(ω))/Im(ε(ω)), (where ε(ω) is the complex dielectric

function of a 50 nm gold film extracted from ellipsometric measurements). In addition to the surface

plasmons of the thin gold field the surface roughness of gold can also couple light to plasmons at

“hotspots” [24]. Interestingly, the plasmon resonance slightly shifts the thermal emission spectra in

vicinity of Au electrodes to a longer wavelength (cf. blue and red curves in figure S9).

Figure S9. Effective spectral emission from hBN/Gr/hBN heterostructure propagating along the

dielectric cavity towards the gold electrodes: black curve – intensity of p-polarised light around

graphene-based nanostructure; red curve – reduced thermal emission in the vicinity of Au electrodes

for the same polarisation; blue curve – plasmon enhancement of the electromagnetic field at the

surface of Au electrodes.

1.0 1.5 2.0 2.5

0.0

0.2

0.4

0.6

0.8

1.0

Em

issio

n (

a.u

.)

Energy (eV)

Au

hBN

hBNGr

SiO2

Si

θ

8. Carriers density: self-gating effect, thermally induced carrier density and disorder

The charge carriers can be distributed non-uniformly due to a disorder, such as electron-hole puddles,

the presence of a temperature gradient and a self-gating at large bias voltages. Compared to SiO2, the

hBN substrate reduces the disorder in graphene to a residual value of ~ 1010 cm−2 [25, 26]. We confirm

the low density of defects and disorder by transport measurements and Raman spectroscopy. For

example, the linewidth of Raman 2D peak was < 20 cm-1 implying mobility > 10000 cm2V-1s-1 at room

temperature [27]. The low-temperature mobility was ~ 100000 cm2V-1s-1 (from transport

measurements). The high mobility can be traced back to a low level of disorder [27, 28]. The most

likely cause of a disorder is strain inhomogeneity on a sub-micron length scale [29]. At the same time,

this low level of disorder does not exclude the possibility of a finite doping due to the presence of

impurities during fabrication procedure as well as exposure to light and high temperatures during device

operation. For example, a laser-induced doping >1012 cm-2 was achieved in hBN/graphene

heterostructures at a low level of disorder (<1011 cm-2) by Neumann et al. in [30]. Nevertheless, although

significant at low temperatures, the effect of disorder completely vanishes above 1000 K (figure S10).

Besides disorder, the self-gating effect leads to non-uniformity of charge distribution, especially at large

bias voltages, Vb. Self-gating effect can be understood as follows: the voltage between the gate plane

and the device channel at the position x is the difference between the local voltage drop at x (Vb·x/L, L

is the channel length) and the voltage applied to the gate electrode Vg. The carrier density induced in

the graphene by the field effect is then 𝑛(𝑥) = (𝑉𝑏∙𝑥

𝐿− 𝑉𝑔) ∙

𝐶𝑔

𝑒, where Cg = εε0/dg is the gate capacitance,

ε and ε0 are relative permittivity and electric constant, dg is the thickness of the gate dielectric, and e is

the elementary charge. The estimated charge carrier density induced by the self-gating effect is plotted

as red dashed line in the figure 3b in the main text. It is interesting to note that self-gating effect

sometimes strongly modifies current-voltage (IV) characteristics of graphene devices with large

transconductance (large gate capacitance) leading to the kink or even negative differential resistance

(NDR) region in IV curve [31]. The presence of NDR strongly depends on the gate capacitance, which

in our case is 30 times smaller, thus preventing us from observing NDR [32, 33]. Furthermore, the

channel length also affects the presence of NDR – the shorter is the channel the more pronounced is the

NDR [34]. Ideally, the length-to-width ratio (L/W) should be below 0.2, while our Hall bar device has

10 times larger L/W, which again prevents us from observing the NDR.

Finally, thermally-induced carrier density scales with temperature as T4, and at temperatures above 1000

K and at realistic gate voltages, the thermally-induced carriers dominate, figure S11. Since the electronic

temperature forms a profile along the device channel (figure S8), the carrier density also varies

correspondingly. This, together with intrinsic doping and bias induced carrier density (self-gating effect)

may lead to the formation of p-n junctions and the asymmetry in current-voltage characteristics and in

the temperature profile with the polarity of applied bias voltage (cf. positive and negative voltage

branches in the figure 3, main text).

Interestingly, in the case of our transport measurements, the cryostat temperature was around 4 K while

the temperature of electron liquid can rapidly reach ~700 K already at low electric fields. The current

densities of 100-200 µA/µm corresponding to low electric fields on Figure 3c are sufficient to raise the

temperature to 500-700 K. For example, using electrical currents of ~100 µA in a similar Hall bar

devices the temperatures in excess of 300K were routinely obtained [35]. Further investigation of this

interesting regime is beyond the scope of this article and comprises a separate study related to the

hydrodynamic behaviour of electron fluid in graphene [36].

Figure S10. Total carrier density (ntotal) vs gate-induced carrier density (ngate) at different levels of

disorder (Δn). Panels (a) to (d) show the effect of temperature smearing out the disorder-induced

carrier density.

(a) (b)

109

1010

1011

1012

1013

109

1010

1011

1012

1013

n (cm-2):

108

109

1010

1011

1012

nto

tal (

cm

-2)

ngate

(cm-2)

T = 4K

(c) (d)

109

1010

1011

1012

1013

109

1010

1011

1012

1013

T = 300K

nto

tal (

cm

-2)

ngate

(cm-2)

n (cm-2):

108

109

1010

1011

1012

109

1010

1011

1012

1013

109

1010

1011

1012

1013

T = 1500K

nto

tal (

cm

-2)

ngate

(cm-2)

n (cm-2):

108

109

1010

1011

1012

109

1010

1011

1012

1013

109

1010

1011

1012

1013

T = 2500K

nto

tal (

cm

-2)

ngate

(cm-2)

n (cm-2):

108

109

1010

1011

1012

Figure S11. Total carrier density (ntotal) vs gate-induced carrier density (ngate) affected by thermally-

induced carriers at different temperatures.

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